We study BPS vortex configurations in three dimensional U(N) Yang-Mills theories with Chern-Simons interaction coupled to scalar fields carrying flavor. We consider two kind of configurations: local vortices (when the number of flavors $N_f=N$), and semi-local vortices (when $N_f>N$). In both cases we carefully analyze the electric and magnetic properties and present explicit numerical solutions.
Various aspects including the construction and the symmetries of Abelian Chern-Simons vortices are reviewed. Extended version of the Lectures delivered at NIKHEF (Amsterdam), July 2006. Typos corrected, some refernces added.
We search for vortices in a generalized Abelian Chern-Simons model with a nonstandard kinetic term. We illustrate our results, plotting and comparing several features of the vortex solution of the generalized model with those of the vortex solution found in the standard Chern-Simons model.
The vortex solutions of various classical planar field theories with (Abelian) Chern-Simons term are reviewed. Relativistic vortices, put forward by Paul and Khare, arise when the Abelian Higgs model is augmented with the Chern-Simons term. Adding a suitable sixth-order potential and turning off the Maxwell term provides us with pure Chern-Simons theory with both topological and non-topological self-dual vortices, as found by Hong-Kim-Pac, and by Jackiw-Lee-Weinberg. The non-relativistic limit of the latter leads to non-topological Jackiw-Pi vortices with a pure fourth-order potential. Explici…
A non-relativistic version of the 2+1 dimensional gauged Chern-Simons O(3) sigma model, augmented by a Maxwell term, is presented and shown to support topologically stable static self-dual vortices. Exactly like their counterparts of the ungauged model, these vortices are shown to exhibit Hall behaviour in their dynamics.
Based on the gauge potential decomposition theory and the $ϕ$-mapping theory, the topological inner structure of the Chern-Simons-Higgs vortex has been showed in detail. The evolution of CSH vortices is studied from the topological properties of the Higgs scalar field. The vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the scalar field $ϕ.$
We study the structure and properties of vortices in a recently proposed Abelian Maxwell-Chern-Simons model in $2 +1 $ dimensions. The model which is described by gauge field interacting with a complex scalar field, includes two parity and time violating terms: the Chern-Simons and the anomalous magnetic terms. Self-dual relativistic vortices are discussed in detail. We also find one dimensional soliton solutions of the domain wall type. The vortices are correctly described by the domain wall solutions in the large flux limit.
We study vortex dynamics in three-dimensional theories with Chern-Simons interactions. The dynamics is governed by motion on the moduli space M in the presence of a magnetic field. For Abelian vortices, the magnetic field is shown to be the Ricci form over M; for non-Abelian vortices, it is the first Chern character of a suitable index bundle. We derive these results by integrating out massive fermions and following the fate of their zero modes.
In this paper we study fractional as well as semi-local Chern-Simons vortices in G = U(1) x SO(2M) and G = U(1) x USp(2M) theories. The master equations are solved numerically using appropriate Ansatze for the moduli matrix field. In the fractional case the vortices are solved in the transverse plane due to the broken axial symmetry of the configurations (i.e. they are non-rotational invariant). It is shown that unless the fractional vortex-centers are all coincident (i.e. local case) the ring-like flux structure, characteristic of Chern-Simons vortices, will become bell-like fluxes - just as …
We discuss a novel method of obtaining the fractional spin of abelian and nonabelian Chern-Simons vortices. This spin is interpreted as the difference between the angular momentum obtained by modifying Schwinger's energy momentum tensor by the Gauss constraint, and the canonical (Noether) angular momentum. It is found to be a boundary term depending only on the gauge field and, hence, is independent of the matter sector to which the Chern-Simons term couples. Addition of the Maxwell term does not alter the fractional spin.
We compute the spin of both the topological and nontopological solitons of the Chern - Simons - Higgs model by using our approach based on constrained analysis. We also propose an extension of our method to the non - relativistic Chern - Simons models. The spin formula for both the relativistic and nonrelativistic theories turn out to be structurally identical. This form invariance manifests the topological origin of the Chern - Simons term responsible for inducing fractional spin. Also, some comparisons with the existing results are done.
We study the locality properties of the vortex operators in compact U(1) Maxwell-Chern-Simons and SU(N) Yang-Mills-Chern-Simons theories in 2+1 dimensions. We find that these theories do admit local vortex operators and thus in the UV regularized versions should contain stable magnetic vortices. In the continuum limit however the energy of these vortex excitations generically is logarithmically UV divergent. Nevertheless the classical analysis shows that at small values of CS coefficient $κ$ the vortices become light. It is conceivable that they in fact become massless and condense due to quan…
We present a (2+1)-dimensional gauged $O(3) σ$-model with an Abelian Chern--Simons term. It shows topologically stable, anyonic vortices as classical solutions. The fields are studied in the case of rotational symmetry and analytic approximations are found for their asymptotic behaviour. The static Euler--Lagrange equations are solved numerically, where particular attention is paid to the dependence of the vortex' properties on the coupling to the gauge field. We compute the vortex mass and charge as a function of this coupling and obtain bound states for two--vortices as well as two--vortices…
We study non-Abelian Chern-Simon BPS-saturated vortices enjoying N=2 supersymmetry in d=2+1 dimensions, with generic gauge groups of the form U(1) x G', with G' being a simple group, allowing for orientational modes in the solutions. We will keep the group as general as possible and utilizing the powerful moduli matrix formalism to provide the moduli spaces of vortices and derive the corresponding master equations. Furthermore, we study numerically the vortices applying a radial Ansatz to solve the obtained master equations and we find especially a splitting of the magnetic fields, when the co…
We have studied BPS vortices in a CPT-odd and Lorentz-violating Maxwell-Chern-Simons-Higgs (MCSH) electrodynamics attained from the dimensional reduction of the Carroll-Field-Jackiw-Higgs model. The Lorentz-violating parameter induces a pronounced behavior at origin (for the magnetic/electric fields and energy density) which is absent in the MCSH vortices. For some combination of the Lorentz-violating coefficients there always exist a sufficiently large winding number $n_{0}$ such that for all $% |n|\geq |n_{0}|$ the magnetic field flips its signal, yielding two well defined regions with oppos…
We study BPS vortices in the mass-deformed non-relativistic ${\C N}=6$ $U(N)_k\times U(N)_{-k}$ Chern-Simons-matter theory. We focus on the massive deformation that preserves the maximal ${\C N}=6$ supersymmetry, and consider a non-relativistic limit that carry 14 supercharges. In this non-relativistic field theory we find Jackiw-Pi type exact vortex solutions combined with $S^3$ fuzzy sphere geometry. We analyse their properties and show that they preserve one dynamical, one conformal and five kinematical supersymmetries among the full super Schrödinger symmetry.
We consider the bosonic sector of a N=2 supersymmetric Chern-Simons-Higgs theory in 2+1 dimensions. The gauge group is U(1)xU(N) and has N_f flavors of fundamental matter fields. The model supports non-Abelian (axially symmetric) vortices when N_f\geq N, which have internal (orientational) moduli. When N_f > N, the solutions acquire additional collective coordinates parameterizing their transverse size. We solve the BPS equations numerically and obtain local (N_f = N) and semi-local (N_f > N) string solutions.
The purpose of this Letter is to present a computation of the interaction energy of gauged O(3) Chern-Simons vortices which are infinitely separated. The results will show the behaviour of the interaction energy as a function of the constant coupling the potential, which measures the relative strength of the matter self-coupling and the electromagnetic coupling. We find that vortices attract each other for $λ> 1 $ and repel when $λ< 1 $. When $λ=1 $ there is a topological lower bound on the energy. It is possible to saturate the bound if the fields satisfy a set of first order partial differen…
In this work we study a nonlinear gauged O(3)-sigma model with both minimal and nonminimal coupling in the covariant derivative. Using an asymmetric scalar potential, the model is found to exhibit both topological and non-topological soliton solutions in the Bogomol'nyi limit.
The non-relativistic Maxwell-Chern-Simons model recently introduced by Manton is shown to admit self-dual vortex solutions with non-zero electric field. The interrelated ``geometric'' and ``hidden'' symmetries are explained. The theory is also extended to (non-relativistic) spinors. A relativistic, self-dual model, whose non-relativistic limit is the Manton model is also presented. The relation to previous work is discussed.
We find self-dual vortex solutions in a Maxwell-Chern-Simons model with anomalous magnetic moment. From a recently developed N=2-supersymmetric extension, we obtain the proper Bogomol'nyi equations together with a Higgs potential allowing both topological and non-topological phases in the theory.
We investigate the vortex-type BPS equations in the ABJM theory without and with mass-deformation. We systematically classify the BPS equations in terms of the number of supersymmetry and the R-symmetries of the undeformed and mass-deformed ABJM theories. For the undeformed case, we analyze the ${\cal N}=2$ BPS equations for U(2)$\times$U(2) gauge symmetry and obtain a coupled differential equation which can be reduced to either Liouville- or Sinh-Gordon-type vortex equations according to the choice of scalar functions. For the mass-deformed case with U($N$)$\times$U($N$) gauge symmetry, we ob…
We study the effect of a Chern-Simons term on the electrically charged and spinning solitons of several $U(1)$ gauged models in $2+1$ dimensions. These are vortices of complex scalar field theories, both with and without symmetry breaking dynamics, and the $O(3)$ Skyrme model. In all cases the gauge decoupling limits are also considered. It is well known that the effect of the Chern-Simons dynamics is to endow vortices with electric charge $Q_e$ and spin $J$, but our main aim here is to reveal a new feature: that the mass-energy $E$ of the electrically charged vortex can be lower than that of …
Making use of $ϕ$-mapping topological current method, we discuss the self-dual vortices in the Abelian Chern-Simons model with two complex scalar fields. For each scalar field, an exact nontrivial equation with a topological term which is missing in many references is derived analytically. The general angular momentum is obtained. The magnetic flux which relates the two scalar fields is calculated. Furthermore, we investigate the vortex evolution processes, and find that because of the present of the vortex molecule, these evolution processes is more complicated than the vortex evolution proce…
In (2+1) dimensions, the Maxwell term $-(1/4) F_{αβ}F^{αβ}$ can be replaced by the Chern-Simons three-form $(κ/4)ε^{αβγ}A_αF_{βγ}$, yielding a novel type of `electromagnetism'. This has been proposed for studying the Quantum Hall Effect as well as High-Temperature Superconductivity. The gauge field can be coupled to a scalar field either relativistically or non-relativistically. In both cases, one admits finite-energy, vortex solutions.
We investigate probe limit vortex solutions of a charged scalar field in Einstein-Maxwell theory in 3+1 dimensions, for an asymptotically AdS Schwarzschild black hole metric with the addition of an axionic coupling to the Maxwell field. We show that the inclusion of such a term, together with a suitable potential for the axion field, can induce an effective Chern-Simons term on the 2+1 dimensional boundary. We obtain numerical solutions of the equations of motion and find Maxwell-Chern-Simons like magnetic vortex configurations, where the magnetic field profile varies with the size of the effe…
We consider a large-N Chern-Simons theory for the attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian, collective-field approach based on the 1/N expansion. We show that the dynamics of density excitations around the ground-state semiclassical configuration is governed by the Calogero or by the Sutherland Hamiltonian, depending on the symmetry of the underlying static-soliton configuration. The relationship between the Chern-Simons coupling constant $ł$ and the Calogero-Sutherland statistical parameter $ł_c$ signalizes some sort of statistical transmutation accompanying the dimensio…
We study ${\cal N}=2$ Chern-Simons-matter theories with gauge group $U_{k_1}(1)\times U_{k_2}(1)$. We find that, when $k_1+k_2=0$, the partition function computed by localization dramatically simplifies and collapses to a single term. We show that the same condition prevents the theory from having supersymmetric vortex configurations. The theories include mass-deformed ABJM theory with $U(1)_{k}\times U_{-k}(1)$ gauge group as a particular case. Similar features are shared by a class of CS-matter theories with gauge group $U_{k_1}(1)\times \cdots \times U_{k_N}(1)$.
The self-duality equations of Chern-Simons Higgs theory in a background curved spacetime are studied by making use of the U(1) gauge potential decomposition theory and $ϕ$-mapping method. The special form of the gauge potential decomposition is obtained directly from the first of the self-duality equations. Using this decomposition, a rigorous proof of magnetic flux quantization in background curved spacetime is given and the unit magnetic flux in curved spacetime is also found . Furthermore, the precise self-dual vortex equation with topological term is obtained, in which the topological term…
Non-Abelian vortices for a supersymmetric {\cal N}=2 Chern-Simons-Higgs theory are explicitly constructed. We introduce N Higgs fields in the fundamental representation of the U(N) gauge group in order to have a color-flavor SU(N) group remaining unbroken in the asymmetric phase. Bogomol'nyi-like first order equations are found and rotationally symmetric solutions are proposed. These solutions are shown to be truly non-Abelian by parameterizing them in terms of orientational collective coordinates. The low energy effective action for the orientational moduli results to be the one-dimensional s…
We examine vortices in Abelian Chern-Simons theory coupled to a relativistic scalar field with a chemical potential for particle number or U(1) charge. The Gauss constraint requires chemical potential for the local symmetry to be accompanied by a constant background charge density/magnetic field. Focussing attention on power law scalar potentials $\sim|Φ|^{2s}$ which do not support vortex configurations in vacuum but do so at finite chemical potential, we numerically study classical vortex solutions for large winding number |n| >> 1. The solutions depending on a single dimensionless parameter …
In this work we consider an Abelian O(3) sigma model coupled nonminimally with a gauge field governed by a Maxwell and Chern-Simons terms. Bogomol'nyi equations are constructed for a specific form of the potential and generic nonminimal coupling constant. Furthermore, topological and nontopological self-dual soliton solutions are obtained for a critical value of the nonminimal coupling constant. Some particular static vortex solutions (topological and nontopological) satisfying the Bogomol'nyi bound are numerically solved and presented.
We investigate the presence of vortex configurations in generalized Maxwell-Chern-Simons models with nonminimal coupling, in which we introduce a function that modifies the dynamical term of the scalar field in the Lagrangian. We first follow a route already considered in previous works to develop the Bogomol'nyi procedure, and, in this context, we use the first order equations to obtain a vortex with a novel behavior at its core. We then go further and introduce a novel procedure to develop the Bogomol'nyi methodology. It supports distinct first order equations, and we then investigate anothe…
Existence of Maxwell-Chern-Simons-Higgs (MCSH) vortices in a Hermitian line bundle $Ł$ over a general compact Riemann surface $Σ$ is proved by a continuation method. The solutions are proved to be smooth both spatially and as functions of the Chern-Simons deformation parameter $κ$, and exist for all $|κ|<κ_*$, where $κ_*$ depends, in principle, on the geometry of $Σ$, the degree $n$ of $Ł$, which may be interpreted as the vortex number, and the vortex positions. A simple upper bound on $κ_*$, depending only on $n$ and the volume of $Σ$, is found. Further, it is proved that a positive {\em lowe…
We have analyzed Maxwell-Chern-Simons-Higgs BPS vortices in a Lorentz-violating CPT-odd context. The Lorentz violation induces profiles with a conical behavior at the origin. For some combination of the coefficients for Lorentz violation there always exists a sufficiently large winding number for which the magnetic field flips its sign.
We consider a generalization of abelian Chern-Simons-Higgs model by introducing a nonstandard kinetic term. In particular we show that the Bogomolnyi equations of the abelian Higgs theory may be obtained, being its solutions Nielsen-Olesen vortices with electric charge. In addition we study the self-duality equations for a generalized non-relativistic Maxwell-Chern-Simons model.
Non Abelian vortices of a SU(2) Chern-Simons--Higgs theory in 2+1 dimensions are constructed numerically. They represent natural counterparts of the U(1) solutions considered by Hong, Kim and Pac, and, by Jackiw and Weinberg. The Abelian embeddings are identified, for all values of the Higgs selfinteraction strength $ν$, resulting in both attractive and repulsive phases. A detailed analysis of the properties of the solutions reveals the existence ofa number of unexpected features. For a certain range of the parameter $ν$, it is shown that the non Abelian vortices have lower energy than their t…
This work introduces a procedure to obtain vortex configurations described by first order equations in generalized Maxwell-Chern-Simons models without the inclusion of a neutral field. The results show that the novel methodology is capable of inducing important modification in the vortex core, leading to vortex configurations with unconventional features.
The dimensional reduction of the three-dimensional fermion-Chern-Simons model (related to Hopf maps) of Adam et el. is shown to be equivalent to (i) either the static, fixed--chirality sector of our non-relativistic spinor-Chern-Simons model in 2+1 dimensions, (ii) or a particular Heisenberg ferromagnet in the plane.
In this work we deal with vortices in Maxwell-Higgs or Chern-Simons-Higgs models that engender long range tails. We find first order differential equations that support minimum energy solutions which solve the equations of motion. In the Maxwell scenario, we work with generalised magnetic permeabilities that lead to vortices described by solutions, magnetic field and energy density with power-law tails that extend farther than the standard exponential ones. We also find a manner to obtain a Chern-Simons model with the same scalar and magnetic field profiles of the Maxwell case. By doing so, we…
We introduce and investigate new models of the Chern-Simons type in the three-dimensional spacetime, focusing on the existence of compact vortices. The models are controlled by potentials driven by a single real parameter that can be used to change the profile of the vortex solutions as they approach their boundary values. One of the models unveils an interesting new behavior, the tendency to make the vortex compact, as the parameter increases to larger and larger values. We also investigate the behavior of the energy density and calculate the total energy numerically.
We study various three-dimensional supersymmetric Maxwell Chern-Simons solitons by using type IIB brane configurations. We give a systematic classification of soliton spectra such as topological BPS vortices and nontopological vortices in $\mathcal{N}=2,3$ supersymmetric Maxwell Chern-Simons system via the branes of type IIB string theory. We identify the brane configurations with the soliton spectra of the field theory and obtain a nice agreement with field theory aspects. We also discuss possible brane constructions for BPS domain wall solutions.
The quantized magnetic flux $Φ=-4πN\sk, N=0,\pm1,...$ of non-topological vortices in the non-relativistic Chern-Simons theory is related to the topological degree of the $S^2\to S^2$ mapping defined by lifting the problem to the Riemann spheres. Regular solutions with finite degree only arise for rational functions, whose topological degree, $N$, is the commun number of their zeros and poles on the Riemann sphere, also called their algebraic order.
We study supersymmetric vortex solutions in three-dimensional abelian gauged supergravity. First, we construct the general U(1)-gauged D=3, N=2 supergravity whose scalar sector is an arbitrary Kahler manifold with U(1) isometry. This construction clarifies the connection between local supersymmetry and the specific forms of some scalar potentials previously found in the literature -- in particular, it provides the locally supersymmetric embedding of the abelian Chern-Simons Higgs model. We show that the Killing spinor equations admit rotationally symmetric vortex solutions with asymptotically …
